 Let $x\in L_1$, and let $y_0\in L_1$ be the first point we choose. Since it doesn't matter where we choose it, because there are no other points in the cloud yet, for simplicity, let $d(x,y_0)=1$. The next point we choose, $y_1$, must be farther away from $x$ than $y_0$, but we can choose it such that $d(x,y_1)<1+\delta$, for any $\delta>0$. So from now on, we will assume that the points we choose are on the sphere $S$ with center $x$ and radius $1$, plus a small $\delta$ for each point. We want to choose the third point in such a way that the forbidden area is as small as possible, so we will place the point somewhere in the circle that is the intersection of the sphere $S$ and the plane that is perpendicular to the segment $(x,y_1)$ and cuts it in half.

In order to minimize the forbidden space, we need to place the new points as close as possible to the points that have already been placed, so we will keep adding points in the intersections of the circles that are the intersection of $S$ and the segments between $x$ and the points we have.